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Mirrors > Home > MPE Home > Th. List > cusgrres | Structured version Visualization version GIF version |
Description: Restricting a complete simple graph. (Contributed by Alexander van der Vekens, 2-Jan-2018.) |
Ref | Expression |
---|---|
cusgrres.v | ⊢ 𝑉 = (Vtx‘𝐺) |
cusgrres.e | ⊢ 𝐸 = (Edg‘𝐺) |
cusgrres.f | ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒} |
cusgrres.s | ⊢ 𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩ |
Ref | Expression |
---|---|
cusgrres | ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ ComplUSGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cusgrusgr 29184 | . . 3 ⊢ (𝐺 ∈ ComplUSGraph → 𝐺 ∈ USGraph) | |
2 | cusgrres.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | cusgrres.e | . . . 4 ⊢ 𝐸 = (Edg‘𝐺) | |
4 | cusgrres.f | . . . 4 ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒} | |
5 | cusgrres.s | . . . 4 ⊢ 𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩ | |
6 | 2, 3, 4, 5 | usgrres1 29080 | . . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ USGraph) |
7 | 1, 6 | sylan 579 | . 2 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ USGraph) |
8 | iscusgr 29183 | . . . 4 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph)) | |
9 | usgrupgr 28950 | . . . . . . . . 9 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UPGraph) | |
10 | 9 | adantr 480 | . . . . . . . 8 ⊢ ((𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph) → 𝐺 ∈ UPGraph) |
11 | 10 | anim1i 614 | . . . . . . 7 ⊢ (((𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph) ∧ 𝑁 ∈ 𝑉) → (𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉)) |
12 | 11 | anim1i 614 | . . . . . 6 ⊢ ((((𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph) ∧ 𝑁 ∈ 𝑉) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁}))) |
13 | 2 | iscplgr 29180 | . . . . . . . . 9 ⊢ (𝐺 ∈ USGraph → (𝐺 ∈ ComplGraph ↔ ∀𝑛 ∈ 𝑉 𝑛 ∈ (UnivVtx‘𝐺))) |
14 | eldifi 4121 | . . . . . . . . . . . . 13 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → 𝑣 ∈ 𝑉) | |
15 | 14 | ad2antll 726 | . . . . . . . . . . . 12 ⊢ ((𝐺 ∈ USGraph ∧ (𝑁 ∈ 𝑉 ∧ 𝑣 ∈ (𝑉 ∖ {𝑁}))) → 𝑣 ∈ 𝑉) |
16 | eleq1w 2810 | . . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑣 → (𝑛 ∈ (UnivVtx‘𝐺) ↔ 𝑣 ∈ (UnivVtx‘𝐺))) | |
17 | 16 | rspcv 3602 | . . . . . . . . . . . 12 ⊢ (𝑣 ∈ 𝑉 → (∀𝑛 ∈ 𝑉 𝑛 ∈ (UnivVtx‘𝐺) → 𝑣 ∈ (UnivVtx‘𝐺))) |
18 | 15, 17 | syl 17 | . . . . . . . . . . 11 ⊢ ((𝐺 ∈ USGraph ∧ (𝑁 ∈ 𝑉 ∧ 𝑣 ∈ (𝑉 ∖ {𝑁}))) → (∀𝑛 ∈ 𝑉 𝑛 ∈ (UnivVtx‘𝐺) → 𝑣 ∈ (UnivVtx‘𝐺))) |
19 | 18 | ex 412 | . . . . . . . . . 10 ⊢ (𝐺 ∈ USGraph → ((𝑁 ∈ 𝑉 ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → (∀𝑛 ∈ 𝑉 𝑛 ∈ (UnivVtx‘𝐺) → 𝑣 ∈ (UnivVtx‘𝐺)))) |
20 | 19 | com23 86 | . . . . . . . . 9 ⊢ (𝐺 ∈ USGraph → (∀𝑛 ∈ 𝑉 𝑛 ∈ (UnivVtx‘𝐺) → ((𝑁 ∈ 𝑉 ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → 𝑣 ∈ (UnivVtx‘𝐺)))) |
21 | 13, 20 | sylbid 239 | . . . . . . . 8 ⊢ (𝐺 ∈ USGraph → (𝐺 ∈ ComplGraph → ((𝑁 ∈ 𝑉 ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → 𝑣 ∈ (UnivVtx‘𝐺)))) |
22 | 21 | imp 406 | . . . . . . 7 ⊢ ((𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph) → ((𝑁 ∈ 𝑉 ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → 𝑣 ∈ (UnivVtx‘𝐺))) |
23 | 22 | impl 455 | . . . . . 6 ⊢ ((((𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph) ∧ 𝑁 ∈ 𝑉) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → 𝑣 ∈ (UnivVtx‘𝐺)) |
24 | 2, 3, 4, 5 | uvtxupgrres 29173 | . . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → (𝑣 ∈ (UnivVtx‘𝐺) → 𝑣 ∈ (UnivVtx‘𝑆))) |
25 | 12, 23, 24 | sylc 65 | . . . . 5 ⊢ ((((𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph) ∧ 𝑁 ∈ 𝑉) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → 𝑣 ∈ (UnivVtx‘𝑆)) |
26 | 25 | ralrimiva 3140 | . . . 4 ⊢ (((𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph) ∧ 𝑁 ∈ 𝑉) → ∀𝑣 ∈ (𝑉 ∖ {𝑁})𝑣 ∈ (UnivVtx‘𝑆)) |
27 | 8, 26 | sylanb 580 | . . 3 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑁 ∈ 𝑉) → ∀𝑣 ∈ (𝑉 ∖ {𝑁})𝑣 ∈ (UnivVtx‘𝑆)) |
28 | opex 5457 | . . . . 5 ⊢ ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩ ∈ V | |
29 | 5, 28 | eqeltri 2823 | . . . 4 ⊢ 𝑆 ∈ V |
30 | 2, 3, 4, 5 | upgrres1lem2 29076 | . . . . . 6 ⊢ (Vtx‘𝑆) = (𝑉 ∖ {𝑁}) |
31 | 30 | eqcomi 2735 | . . . . 5 ⊢ (𝑉 ∖ {𝑁}) = (Vtx‘𝑆) |
32 | 31 | iscplgr 29180 | . . . 4 ⊢ (𝑆 ∈ V → (𝑆 ∈ ComplGraph ↔ ∀𝑣 ∈ (𝑉 ∖ {𝑁})𝑣 ∈ (UnivVtx‘𝑆))) |
33 | 29, 32 | mp1i 13 | . . 3 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑁 ∈ 𝑉) → (𝑆 ∈ ComplGraph ↔ ∀𝑣 ∈ (𝑉 ∖ {𝑁})𝑣 ∈ (UnivVtx‘𝑆))) |
34 | 27, 33 | mpbird 257 | . 2 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ ComplGraph) |
35 | iscusgr 29183 | . 2 ⊢ (𝑆 ∈ ComplUSGraph ↔ (𝑆 ∈ USGraph ∧ 𝑆 ∈ ComplGraph)) | |
36 | 7, 34, 35 | sylanbrc 582 | 1 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ ComplUSGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∉ wnel 3040 ∀wral 3055 {crab 3426 Vcvv 3468 ∖ cdif 3940 {csn 4623 ⟨cop 4629 I cid 5566 ↾ cres 5671 ‘cfv 6537 Vtxcvtx 28764 Edgcedg 28815 UPGraphcupgr 28848 USGraphcusgr 28917 UnivVtxcuvtx 29150 ComplGraphccplgr 29174 ComplUSGraphccusgr 29175 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-oadd 8471 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-dju 9898 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-n0 12477 df-xnn0 12549 df-z 12563 df-uz 12827 df-fz 13491 df-hash 14296 df-vtx 28766 df-iedg 28767 df-edg 28816 df-uhgr 28826 df-upgr 28850 df-umgr 28851 df-uspgr 28918 df-usgr 28919 df-nbgr 29098 df-uvtx 29151 df-cplgr 29176 df-cusgr 29177 |
This theorem is referenced by: cusgrsize 29220 |
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